TREES OF SELF-AVOIDING WALKS. Keywords: Self-avoiding walk, equivalent conductance, random walk on tree

Transcription

1 TREES OF SELF-AVOIDING WALKS VINCENT BEFFARA AND CONG BANG HUYNH Abstract. We consider the biased random walk on a tree constructed from the set of finite self-avoiding walks on a lattice, and use it to construct probability measures on infinite selfavoiding walks. The limit measure (if it exists) obtained when the bias converges to its critical value is conjectured to coincide with the weak limit of the uniform SAW. Along the way, we obtain a criterion for the continuity of the escape probability of a biased random walk on tree as a function of the bias, and show that the collection of escape probability functions for spherically symmetric trees of bounded degree is stable under uniform convergence. Keywords: Self-avoiding walk, equivalent conductance, random walk on tree 1. Introduction An n-step self-avoiding walk (SAW) (or a self-avoiding walk of length n) in a regular lattice L (such as integer lattice Z 2, triangular lattice T, hexagonal lattice, etc) starting at the origin is a nearest neighbor path γ = [γ 0 = 0, γ 1,..., γ n ] that visits no vertex more than once. An infinite self-avoiding walk is a self-avoiding walk of infinite length. Self-avoiding walks were first introduced as a lattice model for polymer chains; while they are very easy to define, they are extremely difficult to analyze rigorously and there are still many basic open questions about them. Let c n be the number of SAWs of length n starting at the origin. The connective constant of L, which we will denote by µ, is defined by c n µ n when n. The existence of the connective constant is easy to establish from the sub-additivity relation c n+m c n c m, from which one can also deduce that c n µ n for all n. Nienhuis [16] gave a prediction that for all regular plan lattices, c n = µ n n α+o(1) where α = 11 32, and this prediction is known to hold under the assumption of the existence of a conformally invariant scaling limit, see e.g. [11]. We are interested in defining a natural probability measure on the set of infinite self-avoiding walks (SAW ) (see the sections 5.2 and 6). Such a measure on the set of the infinite self-avoiding half-plane walks has been constructed already as the weak limit of the uniform measures on the finite self-avoiding walks (see [14]) by using the Kesten s equality [9]. In this paper, we consider a one-parameter family of probability measures on SAW, denoted by (P λ ) λ>λc, defined informally as follows. Denote by H the upper-half plane in Z 2 and by Q the first quadrant; let T Z 2 (resp. T H, T Q, with the appropriate modifications in the definition which we will not specify in what follows) be the tree whose vertices are the finite self-avoiding walks in the plane (respectively half-plane, quadrant), where two such vertices are adjacent when one walk is a one-step extension of the other. We will call this tree the self-avoiding tree on Z 2. Then, consider the continuous-time biased random walk of parameter λ > 0 on T Z 2, which from a given location jumps towards the root with rate 1 and towards each of its children vertices with rate λ. If λ is such that the walk is transient, its path determines an infinite branch in T Z 2 which can be seen as a random infinite self-avoiding walk ω λ ; we will denote by P λ the law of ω λ, omitting the mention of Z2 in the notation, and call it the limit walk with parameter λ. It is well known that there exists a critical value λ c such that if λ > λ c the biased random walk is transient and if λ < λ c it is recurrent. In the general case of biased random walk on a tree, the recurrence or transience of the random walk at the critical point depends in subtle ways on the structure of the tree. The value of λ c on the other hand is easier to determine: 1

2 2 VINCENT BEFFARA AND CONG BANG HUYNH indeed, Lyons [12] proved that it coincides with the reciprocal of the branching rate of the tree. The following proposition give the critical value for self-avoiding trees. Proposition 1.1. Let T Z 2, T H, T Q be defined as above. Then, λ c (T Z 2) = λ c (T H ) = λ c (T Q ) = 1 µ, where µ is the connective constant of lattice Z 2. This is a direct consequence of Proposition 5.10 below. Notice that it is clear from the definition that µ is the growth rate of T Z 2; there are rather large classes of trees, including T Z 2, for which the branching and growth coincide (for instance, this holds for sub- or super-periodic trees, cf. below, or for typical supercritical Galton-Watson trees), but none of the classical results seem to apply to T H or T Q. We now state some properties concerned with the geometry of the limit walk for this family of probability measures. Theorem 1.2. For all λ > λ c, under the P λ measure, the infinite self-avoiding walk (in the plane or half-plane) reaches the line Z {0} infinitely many times almost surely. Theorem 1.3. For all λ > λ c, then P λ (lim sup Rωλ (n) = + ) = 1; P n λ(lim inf n Rω λ (n) = ) = 1. These theorems are proved in Section 6.4. We are mostly interested in the behavior of the limit walk as λ λ c, since this is a natural candidate to be in relation with uniformly sampled long SAWs. We did not quite manage to prove the existence of the limit, but were able to obtain a partial result in this direction by restricting the process to paths formed of bridges of bounded height m, and letting m increase; see Theorem 7.3 for more details. A useful tool in our proofs is the effective conductance of the biased random walk on a tree T, defined as the probability of never returning to the root o of T and denoted by C(λ, T ). Along the way, we will be interested in several properties of it as a function of λ. Most important for us will be the question of continuity: in a general tree, the effective conductance is not necessarily a continuous function of λ. We will derive criteria for continuity, which are forms of uniform transience of the random walk, and apply them to prove that the effective conductance of self-avoiding trees is a continuous function (see Section 5.5): Theorem 1.4. The functions C(λ, T H ) and C(λ, T Z 2) are continuous on (λ c, + ). A related question is that of the convergence of effective conductance along a sequence of trees. More precisely, let (f n ) n denote the effective conductances for a sequence (T n ) of infinite trees, and we assume that (f n ) n converges uniformly towards f 0. The question is: is f the effective conductance of a certain tree? We study this question on a class of particular trees, spherically symmetric trees (recall that T is spherically symmetric if deg x depends only on x, where x denote its distance from the root o and deg x is the number of its neighbors). If S denotes the set of spherically symmetric trees and m N is fixed, define A m := {T S; x T, deg x m} and F m := { f C 0 ([0, 1]) : T A m, C(λ, T ) = f(λ) }. Then (see Section 4.2): Theorem 1.5. Let (f n ) n be a sequence of functions in F m. Assume that f n converges uniformly towards f 0. Then f F m. The paper is structured as follows. In Section 2, we review some basic definitions on graphs, tree, branching number and growth rate of a tree, as well as a few classical results about random walks on trees. Section 3 gathers some relevant examples and counter-examples exhibiting some similarities to the self-avoiding trees while being treatable explicitly. The criterion for the continuity of the effective conductance is given in Section 4. Then Section 5 provides some

3 TREES OF SELF-AVOIDING WALKS 3 background on self-avoiding walks and the self-avoiding trees, and some properties of the limit walks are obtained in Section 6. Finally, we state a few conjectures and conditional results in Section 7. Appendix A isolates the details of an algorithm to improve the readability of the main text. 2. Notation and basic definitions 2.1. Graphs and trees. In this section, we review some basic definitions; we refer the reader to the book [13] for a more developed treatment. A graph is a pair G = (V, E) where V is a set of vertices and E is a symmetric subset of V V, called the edge set, containing no element of the form (x, x). Two vertices are adjacent if their pair that they form is an edge. A path in a graph is a sequence of vertices, any two consecutive of which are adjacent. A simple path is a path which does not pass through any vertex more than once. A graph is connected if, for each pair (x, y) V V, there exist a simple path starting at x and ending at y. A connected graph with no cycles is called a tree. We will always consider trees to be rooted by the choice of a vertex o, called the root. Let T be a rooted tree and x V (T ), the symbol x will denote the height of x, that is the distance from x to o in the graph distance, i.e. the length of the simple path joining o to this vertex; deg x will denote the number of neighbors of x. Let T n be the set of vertices of T with height n. The parent of a vertex is the vertex connected to it on the simple path to the root; every vertex except the root has a unique parent. A child of a vertex v is a vertex of which v is the parent. A vertex is called a leaf if it have no child. We define an order on V (T ) as follows: if x, y V (T ), we say that x y if the simple path joining o to y passes through x. For each x T, we define the sub-tree T x where V (T x ) := {y T : x y} and E(T x ) = E(T ) V (T x ) V (T x ). An infinite simple path starting at o is called a ray. The set of all rays, denoted by T, is called the boundary of T. The set T T can be equipped with a metric that makes it a compact space, see [13] Branching and growth. Definition 2.1. Given a graph G = (V, E) and A, Z two subsets of V, a set Π V is said to separate A and Z (or to be a cut-set between A and B) if every path starting at a point in A and finishing at a point in Z must pass through a vertex of Π. Similarly, if G is infinite and equipped with a marked root o, Π is said to separate o and if every infinite simple path started from o must pass through a vertex of Π; we also call Π a cut-set. For example, let T be a tree, then for all n, T n is a cut-set of T. Definition 2.2. Let T be a tree. The branching number of T is defined by: { br(t ) = sup λ 1 : inf Π where the inf is taken over cut-sets of T. We define also } λ e > 0, e Π gr(t ) = lim sup T n 1/n and gr(t ) = lim inf T n 1/n. In the case gr(t ) = gr(t ), the growth rate of T is defined by their common value and denoted by gr(t ). Proposition 2.3 ([13]). Let T be a tree, then br(t ) gr(t ). In general, the inequality in Proposition 2.3 may be strict: The 1 3 tree (see [13], page 4) is an example for which the branching number is 1 and the growth rate is 2. There are classes of trees however where branching and growth match. Definition 2.4. The tree T is said to be spherically symmetric if deg x depends only on x.

4 4 VINCENT BEFFARA AND CONG BANG HUYNH Theorem 2.5 ([13] page 83). For every spherically symmetric tree T, br(t ) = gr(t ). Definition 2.6. Let N 0: an infinite tree T is said to be N-sub-periodic if for every x T, there exists an adjacency-preserving injection f : T x T f(x) with f(x) N. N-super-periodic if for every x T, there exists an adjacency-preserving injection f : T T f(o) with d(x, f(o)) N. Theorem 2.7 (see [5, 13]). Let T be a tree that is either N-sub-periodic, or N-super-periodic with gr(t ) <. Then the growth rate of T exists and gr(t ) = br(t ) Random walks on trees. Let T be a tree, we now define the discrete-time biased random walk on T. Working in discrete time will make some of the arguments below a little simpler, at the cost of a slightly heavier definition here notice though that the definition of the measure P λ and the main results of the paper are not at all affected by this choice. Let λ > 0: the biased walk RW λ with bias λ on T is the discrete-time Markov chain on the vertex set ot T with transition probabilities given, at a vertex x o with k children, by 1 1+kλ if y is the father of x, p λ (x, y) := λ 1+kλ if y is a child of x, 0 otherwise. If the root has k > 0 children, then p λ (o, x) is 1/k if x is a child of o and 0 otherwise. The degenerate case T = {o} where the root has no child will not occur in our context, so we will silently ignore it. We also allow ourselves to consider the cases λ {0, }, with the natural convention that RW 0 remains stuck at the root. Definition 2.8. Let G = (V, E) be a graph, and c : E R + be labels on the edges, referred to as conductances. Equivalently, one can fix resistances by letting r(e) := 1/c(e). The pair (G, c) is called a network. Given a subset K of V, the restriction of c to the edges joining vertices in K defines the induced sub-network G K. The random walk on the network (G, c) is the discretetime Markov chain on V with transition probabilities proportional to the conductances. Given a network (T, e) on a tree, let π(o) be the sum of the conductances of the edges incident to the root, and denote by o the event that the random walk on (T, e), started at the root, never returns to it. We will write C(o ) := P[o ] and C(o ) := π(o) C(o ). The latter is the equivalent conductance of the network, and its reciprocal R(o ) is the equivalent resistance. The particular case where, on a tree T, for an edge e = (x, y) between a vertex x and any of its children y, c(e) is chosen to be λ x will play a special role, because in that case the random walk on the network is exactly the same process as the random walk RW λ defined earlier. Is this setup, we will denote the equivalent conductance by C(λ, T ) to emphasize its dependency on the parameter λ. Theorem 2.9 (Rayleigh s monotonicity principle [13]). Let T be an infinite tree with two assignments, c and c, of conductances on T with c c (everywhere). Then the equivalent conductances are ordered in the same way: C c (o ) C c (o ). Corollary Let T, T be two infinite trees; we say that T T if there exists an adjacencypreserving injection f : T T. If this holds, then for every λ > 0, C(λ, T ) C(λ, T ). In the case of spherically symmetric trees, the equivalent resistance is explicit: Proposition Let T be spherically symmetric and (c(e)) be conductances that are themselves constant on the levels of T. Then R(o ) = 1 n 1 c, where c n T n n is the conductance of the edges going from level n 1 to level n. Corollary Let T be a spherically symmetric tree. Then RW λ is transient if and only if 1 n λ n T <. n

5 TREES OF SELF-AVOIDING WALKS 5 Theorem 2.13 (Nash-Williams inequality, see [15]). If a and z are distinct vertices in a finite network that are separate by pairwise disjoint cut-sets Π 1, Π 2,..., Π n, then R(a z) This theorem implies the following theorem n ( k=1 e Π k c(e)) 1. Theorem 2.14 (Nash-Williams criterion, see [15]). If Π n is a sequence of pairwise disjoint finite cut-sets in a locally finite network G, then R(o ) 1 c(e). n e Π n In particular, if ( n e Π n c(e) ) 1 = +, then the random walk associated to this family conductances (c(e)) e is recurrent. We end this subsection by stating a classical theorem relating the recurrence or transience of RW λ to the branching of the underlying tree: Theorem 2.15 (see [12]). If λ < 1 br(t ) then RW λ is recurrent, whereas if λ > 1 br(t ), then RW λ is transient. The critical value of biased random walk on T is therefore λ c (T ) := 1 br(t ) The law of first k-steps of the limit walk. Let T be a tree and (c(e)) be conductances on the edges of T such that the associated random walk (X n ) is transient. For every k 0, the walk visits T k finitely many times: we can define an infinite path ω on T by letting ω (k) be the last vertex of T k visited by the walk. Equivalently: (1) ω (k) = x x T i and n 0, n > n 0 : X n T x. Let k N and y 0 = o, y 1, y 2,..., y k be k elements of V (T ) such that (y 0, y 1, y 2,..., y k ) is a simple path: we can then define (2) ϕ c (y 0, y 1, y 2,..., y k ) := P(ω (0) = y 0, ω (1) = y 1,..., ω (k) = y k ). We will refer to this function as the law of first k-steps of limit walk. In the case of the biased walk RW λ, we will denot tho function ϕ λ ; this will not lead to ambiguities. We finish this section with the following lemma. Lemma The value of ϕ c (y 0,..., y k ) depends continuously on any finite collection of the conductances in the network. More precisely, given a finite set U = {e 1,..., e l } of edges and a collection (c(e)) of conductances, let c(u 1,..., u l ) be the family of conductances that coincides with c outside U and takes value u i at e i : then the map is continuous on (R +) l. ψ U,c : (u 1,..., u l ) ϕ c(u1,...,u l )(y 0,..., y k ) 3. A few examples The self-avoiding tree in the plane, which we alluded to in the introduction and will formally introduce in the next section, is sub-periodic but quite inhomogeneous, and the self-avoiding tree in the half-plane sits in none of the classes of trees defined above. To get an intuition of the kind of behavior we should expect or rule out, we gather here a few examples of trees with some atypical features.

6 6 VINCENT BEFFARA AND CONG BANG HUYNH 3.1. Trees with discontinuous conductance. Let 0 < λ 0 1. In the first part of this section, we construct two tree T, T with λ c (T ) = λ c (T ) = λ 0, such that the effective conductances C(λ, T ) and C(λ, T ) of the biased random walk RW λ on T and T satisfy C(λ c (T ), T ) = 0 but C(λ c (T ), T ) > 0. In the second part, we construct a tree T such that C(λ, T ) is not continuous on (λ c, 1). Proposition 3.1. For every x 1, there exist two trees T (x), T (x) such that: br(t (x)) = br(t (x)) = x; RW 1/x is recurrent on T(x) and transient on T (x). Proof. We will construct spherically symmetric trees satisfying both conditions. Denoting by [y] be the integer part of ym first construct the sequence (l i ) i N inductively as follows: [ ] [ ] [ ] x 2 x 3 x n l 1 = [x], l 2 =, l 3 =,..., l n = l 1 l 1 l n 1 2 i=1 l,... i and let T (x) be the tree where vertices at distance i from o have l i children, so that the sizes of the levels of T (x) are given by T n = n i=1 l i. We construct the tree T (x) from the degree sequence (l i ) i N by posing l i = 2l i if i can be written under the form i = k 2, and l i = l i otherwise. Notice that T n = 2 [ n] T n. We first show that both trees have branching number x. Since they are spherically symmetric, it is enough to check that their growth rate is x; the case x = 1 is trivial, so assume x > 1. From the definition, n 1 n x n l i l i x n hence x n x n 1 T n x n i=1 i=1 so gr(t ) = x; the case of T follows directly. The recurrence or transience of the critical random walks can be determined using lemma 2.12: 1 λ n c T n 1 λ n c x n = + so the critical walk on T (x) is recurrent, while for x > 1, 1 λ n c T n 1 = x 1 λ n c (x n x n 1 )2 [ n] x 1 so the critical walk on T (x) is transient. In the case x = 1 one gets 2 [ n] < instead, and the conclusion is the same. Proposition 3.2. For every k N and λ c (0, 1), there exists a tree T with critical drift λ c (T ) = λ c and such that the ratio C(λ)/(λ λ c ) k remains bounded and away from 0 as λ λ + c. Proof. We construct a spherically symmetric tree T which satisfies the conditions of this proposition in a similar way as before. Letting x = 1/λ c > 1, define inductively [ ] [ ] [ ] x 2 x 3 x n l 1 = [x], l 2 = 2 k, l 3 = l 1 3 k,..., l n = l 1 l 2 n k n 1 i=1 l,... i and let T be the spherically symmetric tree with degree sequence (l i ). It is easy to check that br(t ) = x like in the previous proposition; in a similar way, n 1 x n n k i=1 l i n k n i=1 l i x n hence 2 [ n] < x n n k xn 1 (n 1) k T n xn n k. Using Proposition 2.11, the equivalent resistance at parameter λ > λ c is given by R(λ) = 1 λ n T n n k (λx) n C k (λ λ c ) k+1 with a lower bound of the same order but with a different constant, leading to the conclusion.

7 TREES OF SELF-AVOIDING WALKS 7 We end this subsection with the following proposition, showing that discontinuities can occur elsewhere than at λ c : Proposition 3.3. There exists a tree T such that the function C(λ, T ) is not continuous on (λ c, 1). Proof. Let 0 < λ 1 < λ 2 < 1. By proposition 3.1, there exists T a, T b such that λ c (T a ) = λ 1, λ c (T b ) = λ 2 andq C(λ 1, T a ) = 0, C(λ 2, T b ) > 0. We construct a tree T as follows T 1 = {x 1, x 2 } and T x 1 = T a, T x 2 = T b, then λ c (T ) = λ 1. We can see that the function C(λ, T ) is discontinuous at λ 2. Note that continuity properties at λ 1 are actually easier to obtain, and we will investigate them further below The convergence of law of first k-steps. If lim λ λc,λ>λ c C(λ, T ) > 0, by lemma 6.16 the limit of the function ϕ λ,k (y 1,..., y k ) when λ decreases to λ c exists. If lim λ λc,λ>λ c C(λ, T ) = 0, the situation is more delicate and we cannot yet conclude on the limit of the function ϕ λ,k (y 1,..., y k ) when λ decreases to λ c. Indeed, convergence does not always hold, as we will see in a counterexample. The idea of what follows is easy to describe: we are going to construct a very inhomogeneous tree with various subtrees of higher and higher branching numbers, at locations alternating between two halves of the whole tree; a biased random walk will wander until it finds the first such sub-tree inside which it is transient, and escape to infinity within this subtree with high probability. Proposition 3.4. There exists a tree T such that the function ϕ λ,1 (y 0, y 1 ) does not converge as λ λ c. Notation. Let T, T be two trees and A V (T ). We can construct a new tree by grafting a copy of T at all the vertices of A; we will denote this new tree by T A T. Note that for all x A, (T A T ) x T. In the case A = {x}, we will use the simpler notation T x T {x} for T T. Proof. Fix ε > 0 small enough. By Proposition 3.1, for all 0 < a 1, there exists a tree, denoted by T a, such that its branching number is 1 a and C(a, T a) = 0. Let H = Z, seen as a tree rooted at 0, so that the integers is the vertices of H (see the Figure 1). We are going to construct a tree inductively. Let (a i ) i 1 be a decreasing sequence such that a 1 < 1 and denoted a c := lim a i ; assume a c > 0. Choose a sequence b i such that b i (a i+1, a i ) for all i. First, set H 0 := (H 2 Ta1 ) 2 Ta2. We consider the biased random walk RW b1, then it is recurrent on T a1 and transient on T a2. On H 0, the biased random walk RW b1 is transient, and in addition we know that it stays eventually within the copy of T a2. There exists then N 1 > 2 such that the probability that the limit walk remains in that copy after time N 1 1 is greater than 1 ε. Then we set H 1 = (H 0 N 1 Ta3 ). On H 1, the walk of bias b 1 is still transient and still has probability at least 1 ε to escape through the copy of T a2, because T a3 is grafter too far to be relevant. On the other hand, consider the biased random walk RW b2 : it is still transient on H 1 but only through the new copy of T a3. There exists then N 2 > 2 such that the probability that the limit walk remains in that copy after time N 2 1 is greater than 1 ε. We can set H 2 := (H 1 N 2 Ta4 ) and continue this procedure to graft all the trees T ai, further and further from the origin and alternatively on the left and on the right; we denote by H the union of all the H k.

8 8 VINCENT BEFFARA AND CONG BANG HUYNH Figure 1. Tree H It remains to show that the function ϕ λ,1 (0, 1) for the biased random walk on the tree H does not converge. We have br(h ) = max i br(t ai ) = 1 a c and ϕ bi,1 (0, 1) 1 ε if i is odd while ϕ bi,1 (0, 1) 1 ε if i is even. Then, { ϕ bi,1 (0, 1) 1 ε if i = 2k + 1 k 0, ϕ bi,1 (0, 1) ε if i = 2k + 2 This implies that the function ϕ λ,1 (0, 1) does not converge when λ go to a c. The tree we just contructed is taylored to be extremely inhomogeneous. At the other end of the spectrum, some trees have enough structure for all the functions we are considering to be essentially explicit: Definition 3.5. A tree T is called periodic (or finite type ) if, for all v V (T ) \ {o}, there is an adjacency-preserving bijection f : T v T f(v) with f(v) in a fixed, finite neighborhood of the root ot T. Definition 3.6. Let T be a finite tree and L(T ) be the set of leafs of T. We set T 1 L(T ) = T T, 1 T 2 = T 1L(T ) n 1 T,..., T n = T n 1L(T ) T for every n 2. We continue this procedure an infinite number of times to obtain an infinite tree. This infinite tree is called T -finite type and denoted by T,T. Note that T,T is also a periodic tree. Definition 3.7. Let T be a periodic tree and u, v V (T ). We say that u and v have the same type if there is an adjacency-preserving bijection f : T u T v. We denote by type (u, v) := {w V (T v ) : w has the same type with u}. Fact 3.8 (see Lyons [12], theorem 5.1). Let T be a periodic tree and (y 0 = o, y 1, y 2,..., y k ) be a simple path on T. Then the function ϕ λ,k (y 0, y 1,..., y k ) converges when λ decreases to λ c (T ).

9 Moreover, the limit of this function is: TREES OF SELF-AVOIDING WALKS 9 k 1 lim ϕ λ,k (y 0, y 1,..., y k ) = λ λ c i=0 v type (y i,y i+1 ) To keep this paper self-contained, in the rest of this section we provide a proof of a particular case of fact 3.8: Proposition 3.9. Let T be a finite tree and (y 0 = o, y 1, y 2,..., y k ) be a simple path on T,T. Then the function ϕ λ,k (y 0, y 1,..., y k ) of T,T converges when λ decreases to λ c (T,T ). Before showing the proposition 3.9, we study an explicit example of a tree of finite type. Example We define a finite tree T as follows: T 1 = {x 1, x 2 } T 2 = {y} and x 2 is parent of y T n = for all n 3 We can see that T,T is 1-super-periodic. By theorem 2.7, we obtain br(t,t ) = gr(t,t ). Recall that l n is the number of children of a vertex at distance n from o. It is easy to see that l n+1 = l n + l n 1 for all n 1 et l 1 = 2, l 2 = 3. Then we obtain, l n = ( )(1 + This implies that λ c (T ) = We have 5 ( n=1 e Tn 5 2 )n + ( )(1 λ n c ) 1 = n=1 ( ) n l n λ v c. 5 2 )n for all n 1. = +. By theorem 2.14, the biased random walk RW λc(t,t ) on T is recurrent. It remains to show that ϕ λ,1 (o, x 1 ) converges; after the calculations, we obtain ϕ λ,1 (o, x 1 ) = and then ϕ λ,1 (o, x 1 ) converges to λ c when λ go to λ c. λ 1 + 2λ C(λ, T,T ), Lemma Let T be a tree such that deg o = d 0 and { T 1 = {x 1, x 2,..., x d0 } for all t {1, 2,..., d 0 }, λ c (T ) = λ c (T x i ) = λ c and C(λ c, T ) = C(λ c, T x i ) = 0 Then for all i, C(λ, T o, x i ) = C(λ,T 0,x i ) C(λ,T x i) (d xi 1)λ C(λ,T x i) d 0 (1+(d xi 1)λ C(λ,T x i)), where d x i = deg x i. In particular, if converges towards a limit when λ go to λ c, then this limit is equal to (dx i 1)λc d 0. Proof. We can see that where m = C(λ, T o, x i ) = 1 d 0 [ mc + m 2 (1 c)c + m 3 (1 c) 2 c + ] = 1 d 0 mc If C(λ,T 0,x i ) C(λ,T x i) (dx i 1)λ 1+(d xi 1)λ and c = C(λ, T x i ). Then we obtain C(λ, T o, x i ) = converges and as C(λ c, T x i ) = 0, then C(λ,T 0,x i ) C(λ,T x i) (m(1 c)) k, k=0 (d xi 1)λ C(λ,T x i) d 0 (1+(d xi 1)λ C(λ,T x i)). converges towards (dx i 1)λc d 0. Proof of proposition 3.9. First, the biased random walk RW λc on T is recurrent (see [12], theorem 5.1). Recall that L(T ) is the set of all leafs of finite tree T and S i be the set of all

10 10 VINCENT BEFFARA AND CONG BANG HUYNH finite paths starting at origin, ending at one element of L(T ) and pass through x i. We have, for all x L(T ), (T ) x T and we apply several times successive Lemma 3.11 to obtain C(λ, T, x i ) = γ S i f γ 1 (λ)f γ 2 (λ) f γ γ (λ) C(λ, T γ γ ), where f γ i (λ) = m γi λ m γi 1 (1+m γi λc(λ,t γ i)) and m γ i = d γi 1. Since C(λ, T γ γ ) = C(λ, T ), then C(λ, T, x i ) = γ S i f γ 1 (λ)f γ 2 (λ) f γ γ (λ) C(λ, T ). By Lemma 6.16, we obtain ϕ λ,1 (x i ) = C(λ, T, x i ) C(λ, T ) = γ S i f γ 1 (λ)f γ 2 (λ) f γ γ (λ). We observe that for all γ S i, m γ0 decreases towards λ c (T ) and = m(γ γ ), this implies that ϕ λ,1 (x i ) converges when λ (3) lim λ λ c(t ) ϕλ,1 (x i ) = λ c γ. γ S i Remark The equation (3) gives us a way to calculate the critical value of RW λ on T, as the solution of the equation m o x γ = 1. i=1 γ S i 4. The continuity of effective conductance We end the first half of the paper with a few results on the conductance functions of trees, namely we give a criterion for the continuity of C(λ, T ) in λ and study the set of conductance functions of spherically symmetric trees of bounded degree Left- and right-continuity. Lemma 4.1. Let T be a locally finite tree, then C(λ, T ) is right continuous on [0, 1]. Proof. We define S n := inf {k > 0 : d(o, X k ) = n} where X n is RW λ. Then C(λ, T ) = π(o) lim P(S n < S o ). n + We set C(λ, n) := π(o)p(s n < S 0 ). It is easy to see that C(λ, n) C(λ, n + 1). Moreover, by theorem 2.9, C(λ, n) is a continuous increasing function for each n. It implies that C(λ, T ) is the decreasing limit of increasing functions. Therefore C(λ, T ) is right continuous. Definition 4.2. Let T be a locally finite tree. For each x T, we let X x n denote the biased random walk on T x (i.e X x 0 = x, n > 0 : Xx n T x ). We say that T is uniformly transient if λ > λ c, α λ > 0, x T, P( n > 0, X x n x) α λ. It is called weakly uniformly transient if there exists a sequence of finite pairwise disjoint cut-sets Π n, such that λ > λ c, α λ > 0, x + k=1 We can see that if λ c (T ) = 1, then T is uniformly transient. Π k, P( n > 0, X x n x) α λ. Theorem 4.3. Let T be a uniformly transient tree. Then C(λ, T ) is left continuous on (λ c, 1].

11 TREES OF SELF-AVOIDING WALKS 11 Proof. Fix λ 1 > λ c, we will prove that C(λ, T ) is left continuous at λ 1. Choose λ 0 (λ c, λ 1 ). By theorem 2.9, we can find a constant α > 0 (does not depend on λ [λ 0, λ 1 ]) such that λ [λ 0, λ 1 ], x V (T ), P( n > 0, X x n x) α. We give a family of conductances c(e) e E(T ) [0, 1] E, and Y n that is the associated random walk. Let A [0, 1] E be a subset of elements of [0, 1] E such that Y n is transient. Then we define the following function A c(e) e E ψ(c(e) e E ) := C c(e)e E (o ). Recall that T k is the collection of all the vertices at distance k from the root: then we have C(λ, T ) = ψ(λ, λ,... λ, λ 2, λ 2,... λ 2,....). }{{}}{{} T 1 T 2 We will abuse notation until the end of the argument, writing ψ(λ 1, λ 2 2, λ 3 3,...) for ψ(λ 1, λ 1,... λ }{{} 1, λ 2 2, λ 2 2,... λ 2 2,...) }{{} T 1 T 2 so that in particular C(λ, T ) = ψ(λ, λ 2, λ 3,...). Let ε > 0, we choose L N such that (1 α) L < ε. For λ (λ 0, λ 1 ) we have C(λ 1, T ) C(λ, T ) = ψ(λ 1, λ 2 1, λ3 1,...) ψ(λ, λ2, λ 3,...) and by the triangular inequality, we get (4) C(λ 1, T ) C(λ, T ) ψ(λ1,..., λ L 1, b 1 ) ψ(λ,..., λ L, b 1 ) + ψ(λ,..., λ L, b 1 ) ψ(λ,..., λ L, b) where b := (λ L+k ) k 1 and b 1 := (λ L+k 1 ) k 1. Let λ [λ 0, λ 1 ] we denote Sn λ the first hitting point of T n by the random walk with conductances (λ, λ,... λ, λ 2, λ 2,... λ 2,..., λ L, λ L,... λ L, (λ ) L+1,... (λ ) L+1, (λ ) L+2,... (λ ) L+2 ),... }{{}}{{}}{{}}{{}}{{} T 1 T 2 T L T L+1 T L+2 We can see that the law of S λ 1 L and the law of Sλ L are identical. When the random walk reaches T L, it returns to o with a probability strictly smaller than (1 α) L. It implies that (5) ψ(λ,..., λ L, b 1 ) ψ(λ,..., λ L, b) ) 2(1 α) L 2ε. It remains to estimate ψ(λ 1,..., λ L 1, b 1) ψ(λ,..., λ L, b 1 ). By theorem 2.9, we have We apply the lemma 2.16 to obtain ψ(λ 1,..., λ L 1, b 1 ) > 0 and ψ(λ,..., λ L, b) > 0. (6) δ > 0, λ [λ 0, λ 1 ] such that λ 1 λ < δ : ψ(λ 1,..., λ L 1, b 1 ) ψ(λ,..., λ L, b 1 ) < ε. We combine (4), (5) and (6) to get δ > 0, λ [λ 0, λ 1 ] such that λ 1 λ < δ : C(λ 1, T ) C(λ, T ) 3ε. This implies that C(λ, T ) is left continuous at λ 1. In the same method as in the proof of theorem 4.3, we can prove the slightly stronger result (the proof of which we omit): Theorem 4.4. Let T be a weakly uniformly transient tree: then the equivalent conductance C(λ, T ) is left continuous on (λ c, 1].

12 12 VINCENT BEFFARA AND CONG BANG HUYNH 4.2. Conductance functions. Let S denote the set of spherically symmetric trees. Fix m N, and we set A m := {T S; x T, deg x m} and F m := { f C 0 ([0, 1]) : T A m, C(λ, T ) = f(λ) }, F := { f C 0 ([0, 1]) : T, C(λ, T ) = f(λ) }. Definition 4.5. Let T n be a sequence of trees. We say that T n converges locally towards T if k, n 0, n n 0, T n k = T k, where T n is a tree defined by { V (T n ) := {x T, d(0, x) n} E(T n ) = E V (T n ) V (T n ) We are now ready to prove Theorem 1.5. We need the following lemma: Lemma 4.6. Let (f n ) n be a sequence of functions in F m. Assume that f n converges uniformly towards f. Then, there exists a function g F m such that λ, f(λ) g(λ). Proof. Let (T n ) n be a sequence of elements of A m such that n, f n (λ) = C(λ, T n ). Since the degree of vertices of T n are bounded by m, we can apply the diagonal extraction argument. After renumbering indices, there exists a subsequence of (T n ) n, denoted also by (T n ) n, converges locally towards some tree, denote by T. As for all n, T n A m, then T A m. We set g(λ) = C(λ, T ). It remains to show that λ, f(λ) g(λ). Assume that there exists λ 0 such that f(λ 0 ) > g(λ 0 ). We set c = f(λ 0 ) g(λ 0 ) > 0. The sequence f n (λ 0 ) converges towards f(λ 0 ), thus l 1 > 0, n > l 1, f n (λ 0 ) > f(λ 0 ) c 4. Moreover the sequence C n (λ 0, T ) := π(o)p λ0 (S n < S o ) decreases towards g(λ 0 ), It implies that l 2 > 0, n > l 2, C n (λ 0, T ) < g(λ 0 ) + c 4. We take l > 0 such that { fl (λ 0 ) > f(λ 0 ) c 4 C l (λ 0, T ) < g(λ 0 ) + c 4 We have C l (λ 0, T l ) = C l (λ 0, T ), then C l (λ 0, T l ) < g(λ 0 ) + c 4. Moreover, the sequence C k (λ 0, T l ) decreases towards f l (λ 0 ) when k goes to +, thus Then we obtain c < c 4. It is a contradiction. f(λ 0 ) c 4 < f l(λ 0 ) < g(λ 0 ) + c 4. Remark 4.7. The lemma 4.6 is still valid if we take A m := {T T; x T, deg x m} and (f n ) n converges simply to f, where T denote the set of locally finite trees. Proof of theorem 1.5. Fix a diagonal extraction and we take the function g(λ) as in the proof of the lemma 4.6. We will prove that f = g. By lemma 4.6, we have f(λ) g(λ). Assume that there exists λ 0 such that 0 < f(λ 0 ) < g(λ 0 ). We prove that λ < λ 0, f(λ) = 0. By proposition 2.11, if we set β 0 = 1 λ 0, then n, R(λ 0, T n ) = + β0 k R(λ 0, T ) = k=1 k=1 Tk n β0 k Tk

13 TREES OF SELF-AVOIDING WALKS 13 We write As k n, T n k = T k R(λ 0, T n ) =, then k=1 βk 0 T n = βk 0 k k n T n + βk 0 k k>n T n. k R(λ 0, T n ) = k n β k 0 T k + k>n βk 0 Tk n. We know that { limn R(λ 0, T n ) = 1 f(λ 0 ) < lim n R(λ 0, T ) = 1 g(λ 0 ) < 1 f(λ 0 ) We obtain lim n + k>n βk 0 Tk n = 1 f(λ 0 ) 1 g(λ 0 ) > 0. Now we take β > β 0 and we apply the proposition 2.11 in order to get R( 1 β, T n ) = k=0 β k Tk n > k>n It implies that lim n + f n ( 1 β ) = lim n + β k Tk n ( β ) n β 0 k>n βk 0 Tk n 1 = 0. It means R( 1 β,t n ) λ < λ 0, f(λ) = 0. As f 0, we define λ p := inf {0 λ 1 : f(λ) > 0}. We proved that λ > λ p, f(λ) = g(λ). ( β ) n 1 ( β 0 f(λ 0 ) 1 g(λ 0 ) ). As the sequence (f n ) n converges uniformly to f, then f is continuous, and then f(λ p ) = 0. By lemma 4.1, g is right continuous. Then we obtain f(λ p ) = lim λ λ p f(λ) = lim λ λ p g(λ) = g(λ p ) = 0. Moreover f and g are two increasing functions, then λ [0, 1], f(λ) = g(λ). 5. Self-avoiding walks 5.1. Walks and bridges. In this section, we review some definitions on the self-avoiding walk, bridges and connective constant (see [6],[17]). Denote by c n the number of self-avoiding walks of length n, starting at origin on the graph considered. If G is transitive, the sequence c 1/n n converges to a constant when n goes to infinity. This constant is called the connective constant of G. Definition 5.1. An n-step bridge in the plane Z 2 (or half-plane H) is an n-step self-avoiding walk (SAW ) γ such that i = 1, 2,..., n, γ 1 (0) < γ 1 (i) γ 1 (n) where γ 1 (i) is the first coordinate of γ(i). An n-step zero-bridge is an n-step SAW γ such that γ 1 (0) γ 1 (i) γ 1 (n), i = 1, 2,..., n. Let b n denote the number of all n-step bridges with γ(0) = 0. By convention, set b 0 = 1. We have b m+n b m b n, hence we can define Moreover, b n µ n b µn. µ b = lim n + b 1 n = sup b 1 n. n Definition 5.2. An n-step half-space walk is an n-step SAW γ with γ 1 (0) < γ 1 (i), i. We set h n is the number of all n-step half-space walk with γ(0) = 0.

14 14 VINCENT BEFFARA AND CONG BANG HUYNH Definition 5.3. The span of an n-step SAW γ is max γ 1(i) min γ 1(i). 0 i n 0 i n Definition 5.4. Given a bridge (respectively a zero-bridge) γ of length n, γ is called an irreducible bridge (respectively irreducible zero-bridge) if it can not be decomposed into two bridges (respectively zero-bridges) of length strictly smaller than n. It means, we can not find i [1, n 1] such that γ [0,i], γ [i,n] are two bridges (respectively zero-bridges). The set of all irreducible-bridges is denoted by isaw Kesten s and Lawler s measures. For this section, we refer the reader to ([9],[4]) for a more precise description. Denote by SAW the set of all self-avoiding walks on the plane Z 2 or half-plane H. In this part, we review the Kesten measure. He defined a probability measure on the SAW of half-plane from the finite bridges. We let B denote the set of bridges starting at origin and B 0 the set of zero-bridges starting at origin. We let also I denote the set of irreducible bridges starting at origin and I 0 the set of irreducible zero-bridges starting at origin. Let p n denote the number of irreducible bridges starting at origin, of length n and q n denote the number of irreducible zero-bridges starting at origin of length n. We will define a notion of concatenation of paths. If γ 1 = [ [ γ0 1, γ1 1,..., m] γ1 and γ 2 = γ 2 0, γ1 2,..., ] γ2 n are two SAWs, we define γ 1 γ 2 to be the (m + n)-step walk (not necessarily self-avoiding walk) γ 1 γ 2 := [ 0, γ 1 1,..., γ 1 m, γ 1 m + γ 2 1 γ 2 0,..., γ 1 m + γ 2 n γ 2 0]. Similarly, we can define γ 1 γ 2 γ k. We begin with the following equality Fact 5.5 (Kesten [9], Theorem 5). We have n=1 p n µ n = 1. Let us now to define the Kesten measure on the SAW in the half-plane. We fix β 1 µ and let Q β denote the probability measure on I defined by Q β (ω) = β ω Z β, ω I where Z β = ω I β ω. By the fact 5.5 and the remark 5.6 below, Z β is finite and thus Q β is a probability measure on I. Remark 5.6. We have also Λ β < if β < 1 µ. If β > 1 µ then Z β = +, and then Q β can not be defined. If k 1, we consider the product space I k and define the product probability measure Q β k. We write Q β j for an extension to SAW in H as follows, Qβ (ω) = 0 if ω is not of form ω 1 ω 2 ω k and Q β j (H \ Ik ) = 0; Q β j (ω1 ω 2 ω k ) = Q β (ω 1 ) Q β (ω 2 ) Q β (ω k ). We define Q β on I, it is called the β-kesten measure on SAW in half-plane. Fact 5.7. Under the β-kesten measure, the infinite self-avoiding walk, denoted by ω,β K, does not reach the line Ox almost surely. Moreover, if β < 1 µ, we have then P(lim sup n Rω,β K (n) = + ) = 1; P(lim inf n Rω,β K (n) = ) = 1. Now, we define an other probability measure on SAW in half-plane from the finite zerobridges. Then

15 Fact 5.8 (see [4]). For every β (0, 1 µ ], we have TREES OF SELF-AVOIDING WALKS 15 n=0 q n β n < +. In the same way, we can define a probability measure L β on I 0. The infinite half-plane SAW starting at 0 is obtained by choosing ω 1 ω 2..., where ω 1, ω 2,... are independent; ω 1 is chosen from L β ; ω 2, ω 3... are chosen from Q β. The law of infinite half-plane SAW is called β-lawler measure. Fact 5.9. Under the β-lawler measure, the infinite self-avoiding walk, denoted by ω,β L, reaches the line Ox with a probability p which satisfy 0 < p < 1 and it reaches the line Ox a finite number of times almost surely. Moreover, if β < 1 µ, then P(lim sup n Rω,β L (n) = + ) = 1 and P(lim inf n Rω,β L (n) = ) = The self-avoiding tree. Consider the self-avoiding walks in the lattice Z 2 starting at the origin. We construct a tree T Z 2 from these self-avoiding walks: the vertices of T Z 2 are the finite self-avoiding walks and two such vertices joined when one path is an extension by one step of the other. Formally, denote by Ω n the set of self-avoiding walks of length n starting at the origin and V := + n=0 Ω n. Two elements x, y V are adjacent if one path is an extension by one step of the other. We then define T Z 2 = (V, E). We can define with the same way for T H, T Q, where H is a half-plane and Q is a quarter-plane. We know that gr(t Z 2) = br(t Z 2) = µ where µ is the connective constant of lattice Z 2. We calculate the branching number and the growth rate of T H and T Q, that is the contents of the following proposition. Proposition Let T H, T Q be defined as above. Then, gr(t H ) = br(t H ) = gr(t Q ) = br(t Q ) = µ, where µ is the connective constant of the lattice Z 2. Notation. In [9], Kesten proved that all bridges in a half-plane can be decomposed into a sequence of irreducible bridges in a unique way. For every m N, we set: A m := {ω isab, ω m}. An infinite self-avoiding walk is called m-good if it possesses a decomposition into irreducible bridges in A m. We can construct a tree T m from these m-good walks, which we will refer to as the m-good tree. Proof of proposition We know that (see [1], [7]) there exists a constant B and n 0 N such that n > n 0 : c n b n e B n. This implies that gr(t H ) = µ. Let b m n be the number of bridges of length n which possess a decomposition in A m. Then, n, Tn m b m n. We have T m T H, then br(t m ) br(t H ). Moreover, T m is m-super-periodic, so we can apply theorem 2.7 to get br(t m ) = gr(t m ). Then m, br(t H ) br(t m ) = gr(t m ). We will prove that lim gr(t m ) = µ. We have b m+n b m b n, b m n 1 +n 2 b m n 1 b m n 2 and µ = lim n b 1 n. Fix ε > 0, there exists n 0 such that n n 0, µ (b n ) 1 n ε. For each n > n 0, b n = b n n and b kn n (b n n) k = (b n ) k by sub-additivity. It implies that (b n kn ) 1 kn (bn ) 1 n.

16 16 VINCENT BEFFARA AND CONG BANG HUYNH The sequence (b n l ) 1 l increases toward µ n, then (b n kn ) 1 kn µ n gr(t n ). We obtain gr(t n ) k µ ε and then µ gr(t n ) µ ε, n n 0. This implies that lim gr(t n ) = µ and then br(t H ) µ. We apply the proposition 2.3 in order to get br(t H ) gr(t H ) = µ. This implies that br(t H ) = µ and in the same method we obtain gr(t Q ) = br(t Q ) = µ Self-avoiding walks in a wedge. Let C θ be a cone of angle θ of H. Denote by T θ the tree which is constructed from the self-avoiding walks in C θ. Theorem Denote by br(t θ ) (respectively gr(t θ )) the branching number (respectively the growth rate) of T θ. Then br(t θ ) = gr(t θ ) = µ where µ is the connective constant of Z 2. In order to prove this theorem, we have to show the convergence of the connective constants of strips of H with increasing widths and the associated convergence of the branching numbers of the trees constructed from self-avoiding walks of these strips. The convergence of connective constant. Let (B L ) L be the sequence of strips of H where B L is a strip between three lines Iz = 0 and Iz = L. We show the convergence of connective constant of B L towards the connective constant of Z 2. We need the following lemma. Lemma 5.12 (The subadditivity property). Let L, n be two positive natural numbers, we denote by b L n the number of bridges of length n starting at origin in the strip B L. Then, { L, n, m N : b 2L n+m b L mb L n L, n, k N : b 2L kn (bl n) k Remark b kl kn (bl n) k is much easier by sub-additivity. Proof of Lemma We divide the strip B 2L into two small strip of size L, B2L 1, B2 2L (see the figure 1) We denote by S z the symmetry with respect to the line which goes through z and orthogonal to the line Ox. We consider γ 1, γ 2 two bridges in the strip B2L 1 of length m and n, we concatenate γ 1, γ 2, we obtain a path γ 12 := γ 1 γ 2. We can see thatγ 12 is a bridge of B 2L of length m + n. This implies that for all L, n, m N, b 2L n+m b L mb L n. Next, if one takes the third bridge γ 3 of B 1 2L of length p, we concatenate γ 1, γ 2, γ 3 as follows. { γ 123 = γ 12 γ 3 if R(γ 12 ( γ 12 )) L γ 123 = γ 12 S γ12 ( γ 12 )(γ 3 ) if R(γ 12 ( γ 12 )) > L We can see that γ 123 is a bridge of B 2L of length m + n + p. If we take m = n = p, then b 2L 3n (bl n) 3. We repeat the same strategy to obtain the result of lemma Proposition We denote by µ L the connective constant of the strip B L, then lim L µ L = µ where µ is the connective constant of Z 2. Proof. We define b Q n the number of bridges of Q of length n, then L : lim n (b L n) 1 n = µ L L : b L L = bq L lim n (b Q n ) 1 n = µ L, n, k : b 2L kn (bl n) k From these relations, we prove that lim L µ L = µ. Let ε > 0, we take n 0 such that: (b Q n ) 1 n µ ε, n n 0. The sequence (b 2n 0 kn 0 converges towards µ 2n0. Moreover b 2n 0 kn 0 ) 1 (b n 0 n 0 ) k, thus (b 2n 0 kn 0 (b n 0 n 0. By making k tend to +, we obtain µ 2n0 µ ε. As the sequence (µ L ) L is an increasing sequence, thus L > 2n 0 : µ L > µ ε. This implies that the sequence µ L converge towards µ when L goes to +. kn 0 ) 1 n 0 ) 1 kn 0

17 TREES OF SELF-AVOIDING WALKS 17 Figure 2. A concatenation of 3 bridges in B 1 2L. Proposition lim L br(t BL ) = µ. Proof. Recall the definition of A m in the proof of Proposition An infinite self-avoiding walk of B L is called m-good walk if it possesses a decomposition into irreducible bridges in A m. We construct a tree TB m L from these m-good walks. We set b L,m n be the number of bridges of B L of length n which possess a decomposition in A m. We know that all bridges in a half-plane can decompose into a sequence of irreducible bridges of the unique way. This implies that all bridges in the strip B L can decompose into a sequence of irreducible bridges of the unique way. Then for all n, (TB m L ) n b L,m n. { L, n, k : b 2L nk We obtain, (bl n) k L, m, n, k : b 2L,m nk (b L,m n ) k Let ε > 0, we apply the proposition 5.14, there exists L 0 such that: µ µ L0 > µ ε.

18 18 VINCENT BEFFARA AND CONG BANG HUYNH Moreover, µ L0 = lim(b L 0 n ) 1 n, there is also m: (b L 0 m ) 1 m > µ L0 ε. Then, (b 2L 0,m km ) 1 km (b L 0,m m ) 1 m = (b L 0 m ) 1 m µl0 ε µ 2ε. We obtain gr(tb m 2L0 ) µ 2ε. As TB m 2L0 is (m + 2L 0 )-super-periodic and gr(tb m 2L0 ) < +, we apply the theorem 2.7 to get gr(tb m 2L0 ) exists and gr(tb m 2L0 ) = br(tb m 2L0 ). Moreover TB m 2L0 T B2L0, then br(t B2L0 ) µ 2ε. The sequence br(t BL ) is an increasing sequence, we obtain: L 2L 0 : br(t BL ) µ 2ε. Moreover br(t BL ) µ because T BL T H. This implies that lim L br(t BL ) = µ. Proof of Theorem Let ε > 0, we apply the proposition 5.14 and 5.15, there exists a constant L such that: br(t BL ) > µ ε and gr(t BL ) > µ ε. We take N such that: N tan θ > L. We have the cone C θ contains a strip of size greater than L, origin at (N tan θ, 0). This implies that br(t θ ) > µ ε; gr(t θ ) > µ ε. As ε is arbitrary, then br(t θ ) = µ and gr(t θ ) = µ Continuity of C on T H. Now, we apply the results in Section 4.1 for the self-avoiding trees T H and T Z 2. Theorem The function C(λ, T H ) (or C(λ, T Z 2)) is continuous on ( 1 µ, 1), where µ is the connective constant of the lattice Z 2. Proof. The right continuity of C(λ, T H ) is a consequence of the lemma 4.1. In order to prove the left continuous, we seek to apply the theorem 4.4. For this, we prove that T H is weakly uniformly transient. In the half-plane H, we define a sequence of rectangles (R n ) n 1 where R n is the rectangle with 4 vertices ( n, 0); ( n, n); (n, n); (n, 0). We define a sequence of pairwise disjoint cut-sets from these rectangles as follows: } Π n := {γ : γ( γ ) R n and k < γ, γ(k) R o n. We set Γ := Π n. It remains to verify that λ > λ c (= 1 µ ), α λ > 0, x Γ, P( n > 0, X x n x) α λ. Recall that T Q denote the self-avoiding tree from a quarter-plane. We can see that, for every x Γ, T x contains the tree T H or T Q. We conclude thank to the proposition Remark The self-avoiding trees T H and T Z 2 are not uniformly transient. Recall that B n is a strip of H and T Bn is the self-avoiding tree which is constructed from self-avoiding walks in B n. Let f n (λ) := C(λ, T Bn ). Theorem The sequence of functions (f n ) n converges uniformly towards a continuous function f if and only if RW λc on T H is recurrent, where 1 λ c = µ is the connective constant of lattice Z 2. In order to prove the theorem 5.18, we need the following lemma. Lemma For all k, f k ( 1 µ ) = 0. Proof. We use the theorem 2.14 for this proof. We fix k N and for each n, we define the rectangle R n with 4 vertices ( n, 0); ( n, k); (n, k); (n, 0). We define a sequence of cut-sets from these rectangles as follows Π n := {γ : R(γ( γ )) { n, n} ; k < n, R(γ(k) < n and 0 I(γ(k)) k}.

19 TREES OF SELF-AVOIDING WALKS 19 We will estimate ( e Π n c(e) ) 1. We can see that a self-avoiding walk γ Πn of length i can expand into a bridge of length i + k + 1 and as γ Π n then n γ 2kn, it implies that and then It means RW 1 µ e Π n c(e) ( n 2kn i=n e Π n c(e) b i+k+1 2kn µ i 3 k ) 1 n i=n b i µ i 2k3k n, 1 2k3 k n = +. on T Bk is recurrent and then f k ( 1 µ ) = 0. Proof of Theorem Assume that (f n ) n converges towards a continuous function f. We will prove that RW ( 1 µ, T H) is recurrent. We set g(λ) = C(λ, T H ), we prove that λ > 1 µ, f(λ) = g(λ). We fix λ > 1 µ and define a sequence of pairwise disjoint cut-sets O n by considering the first time reaches the rectangles (as in the proof of the theorem 5.16). Let T be an arbitrary tree and C denote its cut set. We set T (0 C) := {x T : y C, x y}. We can see that n, T Bn (0 O n ) = T H (0 O n ). We proved that lim λ c (T Bn ) = 1 µ, then we can find l > 0 such that λ c(t Bl ) < λ. Then we set Then, there exists k > 0 such that We obtain n > k, m = C(λ, T Bl ) > 0. { Pλ (0 O n ) > g(λ) ε (1 m) n < ε i > k + l, f i (λ) > g(λ) 2ε Then g(λ) f(λ) > g(λ) 2ε. This implies that f(λ) = g(λ). Since f is a continuous function, thus f( 1 µ ) = lim f(λ) = lim g(λ) = g( 1 λ 1 λ 1 µ ). µ µ By lemma 5.19, we have f( 1 µ ), and then RW 1 on T H is recurrent. µ Conversely, if RW 1 on T H is recurrent, it is easy to see that µ λ [0, 1], f(λ) = g(λ). By theorem 5.16 and moreover g( 1 µ ) = 0, we have then g is continuous function and then f is continuous. In the same way, we can prove the following: Proposition With the same notations as in the proof of the proposition 5.10, set f n (λ) := C(λ, T n ). Then the sequence of functions (f n ) n converges towards a continuous function f if and only if RW λc on T H is recurrent, where 1 λ c = µ is the connective constant of lattice Z The biased walk on the self-avoiding tree We now begin the study of our main object of interest, which is the biased random walk on the self-avoiding tree. We will use the results obtained in the previous section to prove properties of the limit walk; in the next section, we will gather a few natural conjectures.